6.19.69 problem section 9.3, problem 69

Internal problem ID [2216]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 69
Date solved : Tuesday, September 30, 2025 at 05:25:06 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }-5 y^{\prime }+6 y&=2 \,{\mathrm e}^{x} \left (1-6 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=7 \\ y^{\prime \prime }\left (0\right )&=9 \\ \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 25

ode:=diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-5*diff(y(x),x)+6*y(x) = 2*exp(x)*(1-6*x); 
ic:=[y(0) = 2, D(y)(0) = 7, (D@@2)(y)(0) = 9]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x}-{\mathrm e}^{-2 x}+{\mathrm e}^{3 x} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 27

ode=D[y[x],{x,3}]-2*D[y[x],{x,2}]-5*D[y[x],x]+6*y[x]==2*Exp[x]*(1-6*x); 
ic={y[0]==2,Derivative[1][y][0] ==7,Derivative[2][y][0] ==9}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (x^2+2\right )-e^{-2 x}+e^{3 x} \end{align*}

✓ Sympy. Time used: 0.196 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((12*x - 2)*exp(x) + 6*y(x) - 5*Derivative(y(x), x) - 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 7, Subs(Derivative(y(x), (x, 2)), x, 0): 9} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x^{2} + 2\right ) e^{x} + e^{3 x} - e^{- 2 x} \]